Fix #3998 and #3992 (#4000)

Fixes #3998
Fixes #3992

code/modules/mob/living/silicon/ai/ai.dm

@@ -279,6 +279,12 @@ var/list/ai_verbs_default = list(
 		aiPDA.owner = pickedName
 		aiPDA.name = pickedName + " (" + aiPDA.ownjob + ")"
 
+	//Set the ID Name
+	if(idcard)
+		idcard.registered_name = pickedName
+		idcard.assignment = "AI"
+		idcard.update_name()
+
 	setup_icon() //this is because the ai custom name is related to the ai name, so, we just call the setup icon after someone named their ai
 
 /*

code/modules/mob/living/silicon/robot/robot.dm

@@ -336,6 +336,12 @@
 	// if we've changed our name, we also need to update the display name for our PDA
 	setup_PDA()
 
+	// We also need to update our internal ID
+	if(idcard)
+		idcard.assignment = prefix
+		idcard.registered_name = changed_name
+		idcard.update_name()
+
 	//We also need to update name of internal camera.
 	if (camera)
 		camera.c_tag = changed_name